dynamic modeling and simulation of spatial manipulator with flexible links and joints

8/11/2019 Dynamic Modeling and Simulation of Spatial Manipulator With Flexible Links and Joints

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11th International Conference on Vibration Problems

Z. Dimitrovovaet.al. (eds.)

Lisbon, Portugal, 912 September 2013

DYNAMIC MODELING AND SIMULATION OF SPATIAL

MANIPULATOR WITH FLEXIBLE LINKS AND JOINTS

Rajesh K Moolam*1, Francesco Braghin1, Federico Vicentini2

1Politecnico di Milano, Milan, Italy{rajeshkumar.moolam,francesco.braghin}@polimi.it

2ITIA-CNR, Milan, Italy

[email protected]

Keywords: Spatial Flexible Manipulators, Flexible Links, Flexible Joints, Dynamic Modeling.

Abstract.Dynamic modeling of the robot manipulator is very important to design model-based

control and for simulation purposes. A steady increase in the demand of high speed operations,

low energy consumption and increase in payload to arm weight ratio motivates the use of light

weight materials to build manipulators. With the use of light weight material, the rigid link

assumption is no longer valid and, in some cases, the transmission system at manipulator joints

can introduce flexibility. Ignoring the link and joint flexibility can cause poor estimation of

dynamic parameters and, eventually, poor performance of the control design. To obtain an

accurate dynamic model of a robot manipulator, the flexibility of the links and joints should

be accounted. The inclusion of the dynamics due to flexibility makes the robot manipulator a

continuous system and requires infinite degrees of freedom to estimate the dynamic parameters.

It also establishes strong coupling between gross rigid body motion and elastic deformations of

links and joints.

In this paper, a general purpose algorithm is presented that allows to obtain a dynamic

model of spatial flexible manipulators for model based control design and simulation purposes.

Both link and joint flexibilities are considered in the dynamic modeling. The flexible links are

discretized to get a finite dimensional dynamic model. The deformation of each link is assumed

to be due to both bending and torsion. The deformation of the joints is assumed to be due to puretorsion. The deformation of each link is assumed to be small relative to the rigid body motion.

Thus, the configuration of each link is defined as the sum of rigid and elastic coordinates using a

floating reference frame. The dynamic model is first derived using the principle of virtual work

along with finite element method in generalized coordinates for general purpose implementa-

tion. Then, the system of equations in generalized coordinates is converted into independent

coordinate form using a recursive kinematic formulation based on the topology of a manipula-

tor. The advantage of general purpose algorithm is it uses minimum set of equations that define

the dynamics of flexible manipulator, which is required in control design to reduce computation

cost. In addition, it allows the dynamic modeling of any arbitrary manipulator configuration.

Numerical simulation results of an open chain RRR manipulator with flexible links and flexiblejoints is presented to show the effects of flexibility on robot manipulator dynamics.


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Rajesh K Moolam, Francesco Braghin, Federico Vicentini

Figure 1: Arbitrary point on a flexible link. Figure 2: Elastic coordinates on finite element.

expressed in a tangential local floating frame. The constrained equations due to the connectivityof each link are added to equations of motion by using Lagrangian multiplier. The redundant

dynamic model requires more computational power. In [13] develops a recursive Newton-Euler

formulation for a flexible link open loop robot system. The recursive formulation converts the

redundant form into minimum set of equations. However, this formulation ignores the joint

flexibility. Ignoring the joint flexibility can significantly affect the system Eigen frequencies

[14].

In this paper, the systematic approach for the dynamic modeling of spatial flexible manipu-

lators considering both link and joint flexibility is presented. The link flexibility is assumed due

to bending and torsion, where shear deformations are neglected. The joint flexibility is assumed

due to pure torsion. The deformation of each link is assumed to be small with respect to gross

rigid body motions. Hence, the configuration of each link is defined as the sum of rigid and elas-

tic coordinates using floating reference frame. A general purpose program is developed based

on the principle of virtual work and finite element method. The dynamic model is first derived

using generalized coordinates for general purpose implementation. Then, a recursive kinematic

formulation is presented based on the topology of the flexible manipulator which converts the

system of equations in absolute coordinates into relative coordinates.

2 Kinematics of Flexible Manipulators

The kinematic equations that define an arbitrary displacement of a flexible link i shown

in Figure 1 is derived using floating reference frame formulation. Floating reference frame

formulation uses two sets of coordinates i.e., body reference and elastic coordinates. The body

reference coordinates describe the position and orientation of body reference frame XiYiZiwithrespect to global coordinate system XY Z. The elastic coordinates describe the deformations onflexible linkiwith respect to body reference frame XiYiZi. The elastic deformations on flexiblelink are approximated using finite element method to obtain finite set of elastic coordinates. The

elastic coordinates of finite element shown in Figure2is defined using floating reference frame

XijYijZij with respect to body reference frameXiYiZi. The position of an arbitrary point onflexible linki can be defined as

ri=Ri+ Aiui (1)

whereRiis the position vector of body reference frameXiYiZi.Aiis the transformation matrixdefined using euler parametersi = [0 1 2 3]. ui is the local position vector defined with

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respect toXiYiZi.

ui=uri + u

ei (2)

whereuri

is the undeformed position vector anduei

is deformation vector which is defined as

uei =Siqei (3)

In which Si is the shape function matrix and qei is elastic coordinates vector. Differentiating

Eq.(1) with respect to time gives the velocity of an arbitrary point on a flexible link. It is written

as

ri= Ri+ Ai(iui) +AiSiqei (4)

wherei is angular velocity vector defined in body coordinate system XiYiZi.

3 Dynamic Modeling of Flexible Manipulators

The equations of motion of a flexible manipulator are derived using the principle of virtual

work. The dynamics of flexible links is first derived using absolute coordinates for the general

purpose implementation. Then the system of equations in absolute coordinates is converted into

minimum set of equations or relative coordinate form using recursive kinematic equations.

3.1 Flexible link modeling

The virtual work of total forces acting on flexible linki is defined as

Wi= Wi

i+W

i

s +Wi

e (5)

whereWii , Wsi, and W

ei are respectively the virtual work of inertia forces, elastic forces,

and external forces. The flexible linki is discretized using finite element method to get finite

dimensional dynamic model. The representation of finite elementij on linki is shown in Fig-

ure2. The virtual work of flexible linki can be obtained by summing up the virtual work of its

elements.

The virtual work of inertia forces acting on element ij is written as

Wiij =Vij

ij rTij rij dVij (6)

whereij andVij are respectively, the mass density and volume of element ij. rij andrij arerespectively the acceleration vector and virtual displacements of an arbitrary point on element

ij. The virtual displacementrij is written as

rij =Lijqij (7)

where

Lij =

I AiuijGi AiSij , qij = Rij ij qeij T (8)whereqij is generalized coordinates of element ij. The acceleration vector rij of an arbitrarypoint can be obtained by differentiating Eq.4with respect to time.

rij =Lij qij+Qij (9)

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in whichqij is the generalized accelerations andQij is the quadratic term which is written as

Qij = Aii2 uij+ 2AiiSijqeij (10)

Substituting acceleration vectorrij and virtual displacementsrij in Eq.6gives

Wiij =Vij

ij qTij L

Tij Lijqij dVij+

Vij

ij QTij Lijqij dVij (11)

Wiij =qTij MijQ

vT

ij

qij (12)

whereMij andQvij are respectively the inertia matrix and quadratic velocity term.

Mij =Vij

ij

I Aiuij Gi AiSij

GT

i

uT

ij

uijGi G

T

i

uT

ijSij

symmetric S

T

ijSij

dVij (13)

and

Qvij =Vij

ij

IGTi uTijATiSTijA

Ti

Qij dVij (14)The virtual work of elastic forces due to the deformation of element ij can be defined as

Wsij =Vij

TijijdVij (15)

whereij andij are stress and strain vectors of element ij.

ij =Dijueij =DijSijq

eij (16)

ij =Eijij =EijDijSijqeij (17)

Substituting Eq.16and Eq.17in Eq.15gives

Wsij =qeT

ij

Vij

(DijSij)TEijDijSijdVij

qeij =q

eT

ij Keijq

eij (18)

whereDij is the differential operator, Sij is the element shape function matrix and Eij is theelastic coefficient. The virtual work of external forces acting on elementij is defined as

Weij =QeT

ij qi (19)

SubstitutingWiij ,Wsij andW

eij in Eq.5yields

Wij =qTij MijQ

vT

ij qeT

ij KeijQ

eT

ij

qij (20)

From the Eq.20the equations of motion can be rearranged as

Mij qij =Qeij+ Q

vij+ Q

sij (21)

whereQeij are the external forces applied. Qvij andQ

sij are respectively the quadratic veloc-

ity term and elastic forces. Eq.21can be extended to all finite elements in flexible link andassembled based on element connectivity to form a dynamic model of flexible link.

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Figure 3: Flexible jointj assembly

3.2 Flexible joint modeling

The revolute jointjwith actuator and transmission system is shown in Figure3.The actuator

is assumed as electric motor and torsional spring represents the flexibility induced due to trans-

mission system.j andi respectively the rotations of actuatorj and linki. The virtual work oftorque exserted on linki by actuatorj and transmission system is defined as

Wj = (Jj j+ Cj(j i) +Kj(ji)Tj)ij (22)

whereJj is the inertia of the motor.Kj andCj are the stiffness and damping coefficients oftransmission system.Tj is the torque produced by motor. ij is the virtual change at joint. The

equations of motions of joint assembly is written as

Jj j+ Cj(j i) +Kj(ji) =Tj (23)

3.3 Recursive Kinematic Formulation

Consider two flexible link i-1 and i shown in Figure 4which are connected by a revolute

jointj. The joint allows relative rotation along joint axis and has one rigid body coordinates i.The following kinematic relationship for revolute joint holds the relation between generalized

Figure 4: Representation of Relative body Coordinates

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coordinates and joint coordinates

Ri+Aiuji Ri1Ai1u

ji1= 0 (24)

i=i1+ ji1ji + i,i1 (25)

whereuji anduji1 are local position vectors of joint defined on linkiand i-1respectively.

ji

andji1 are respectively the local angular velocity vectors of joint due to elastic deformationson linki andi-1. These vector quantities are defined as

ji1=Ai1Sjri1q

ei1 (26)

ji =AiSjri q

ei (27)

In whichSjri andSjri1are respectively the constant shape function matrix of joint rotations due

to elastic deformations on linki and i-1. i,i1 is relative angular velocity vector of linki withrespect to linki-1 is expressed as

i,i1= i1i (28)

wherei1 is rotation axis defined with respect to linki-1 in global coordinate systemX Y Z.

i1= Ai1i1 (29)

i1is constant vector defined with respect to linki-1in body coordinate system Xi1Yi1Zi1.

Differentiating Eq.24twice with respect to time and Eq.25once with respect to time gives

RiAiuji Gii+AiSjti qei = Ri1Ai1uji1Gi1i1+ Ai1Sjti1qei1+ R (30)i= i1+ Ai1S

jri1q

ei1AiS

jri q

ei + Ai1i1i+ (31)

whereSjti andSjti1 are respectively the shape functions of joint translations defined on linki

andi-1.Rand are written as

R =Ai i2

uji +Ai1 i12

uji12AiiSjti qei + 2Ai1i1Sjti1qei1 (32) =Ai1i1i1i+ Ai1i1Sjri1qei1AiiSjri qei (33)

The equation Eq.30and Eq.31can be written in a compact form as

Diqi = Di1qi1+ HiPi+i (34)

where

Di = I Ai

uj

i Gi AiSjti

0 AiGi AiSjri

0 0 I (35)

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Di1=

I Ai1uj

i1Gi1 Ai1Sjti1

0 Ai1Gi1 Ai1Sjri1

0 0 0

(36)

Hi= Ai1i1 0

0 I (37)

Pi=

i qei

T(38)

i =

R T

(39)

The generalization of recursive formulation for an link manipulator is expressed as

D1 0 0

D1 D2 0... ...

. . . 00 0 Dn1 Dn

q1

q2...

qn

= H1 0 0

0 H1 0... ...

. . . 00 0 0 Hn

P1

P2...

Pn

+ 1

2...

n

(40)

The generalized accelerations qi in absolute coordinates can be expressed in terms of relativecoordinates as

qi = BiPi+i (41)

where

Bi =

D1 0 0

D1 D2 0...

... . . . 0

0 0 Dn1 Dn

1

H1 0 0

0 H1 0...

... . . . 0

0 0 0 Hn

(42)And

i=

D1 0 0D1 D2 0

... ...

. . . 00 0 Dn1 Dn

1

12...

n

(43)

Substituting the generalized acceleration qi in Eq. 21and premultiplying with BTi gives thedynamic model of flexible links in relative coordinates form.

BTi MiBiPi=B

Ti (Q

ei + Q

vi +Q

si Mii) (44)

which can written as

MiPi= Qi (45)

where

Mi=BTi MiBi,

Qi=BTi (Q

ei + Q

vi + Q

si Mii) (46)

The Eq.45along with Eq.23 provides a coupled and nonlinear dynamic model of flexiblelink and flexible joint which can be used for simulation and model based control design purpose.

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Figure 5: Spatial RRR flexible manipulator.

4 Numerical Simulation

A spatial RRR manipulator shown in Figure5, is considered to demonstrate the effect of link

and joint flexibility on manipulator dynamics. Each flexible link is discretized using two finiteelement beams with six degrees of freedom on each node and one degree of freedom for rigid

body rotations i.e. i wherei = 1, 2, 3. Manipulator joint has one rigid body rotation i.e. jwherej = 1, 2, 3. The torsional stiffnessKj forj = 1, 2, 3at manipulator joints is defined as5000 Nm/rad. The damping effects on links and joints are ignored. The physical parameters of a

RRR spatial manipulator is presented in Table1. Uniform cross-section and material properties

are assumed on each link. The numerical simulation for different cases considering rigid links

and rigid joints, flexible links and rigid joints, flexible links and flexible joints is performed

to study the effect of flexibility on manipulator dynamics. A constant torque of 400 Nm, is

applied at manipulator joints for each case to compare manipulator motion. The deformations

of manipulator endeffector X4Y4Z4along the X,Y and Z direction is shown in Figure6,Figure7and Figure8 respectively. It shows the deformations of flexible manipulator endeffector with

respect to the rigid manipulator endeffector motion. The deformation of flexible manipulator

joint i with respect to rigid manipulator joint motion is shown in Figure 9, Figure 10 andFigure11respectively. The numerical simulation results show significant effect of link and joint

flexibility on the overall manipulator motion. In addition, the joint flexibility in manipulator can

significantly alter the motion of the manipulator.

Parameter Link 1 Link 2 Link 3

Link Length (m) 1 4.0 3.5

C/s Area (m2) 0.028 0.0020 0.0008Moment of Inertia (m4) 8.33107 6.24107 5.37107

Polar moment of Inertia (m4) 1.66106 1.24106 1.07106

Tensile Modulus (MPa) 206000

Shear Modulus (MPa) 79300

Density (Kg/m3) 8253

Table 1: The Physical Parameters of a RRR flexible manipulator.

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Figure 6: Deformation of flexible manipulator endeffector with respect to rigid manipulator endeffector motion in

X-direction.


Y-direction.


Z-direction.

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Figure 9: Deformations of flexible manipulator joint 1with respect to rigid manipulator joint motion.



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5 Conclusions

The light weight manipulators have many advantages over rigid manipulators. However,

the low stiffness of links and in some cases the transmission systems at manipulator joints can

introduce flexibility in the light weight manipulators. The effect of link and joint flexibility on

manipulator dynamics is studied using a spatial RRR manipulator. The numerical simulation

of different cases considering rigid links and rigid joints, flexible links and rigid joints, flexible

links and flexible joints is performed. The simulation results show significant effect of flexibility

on the overall manipulator motion. An accurate dynamic model that incorporates both link and

joint flexibility is mandatory for model based control design. A general purpose algorithm

that allows to obtain an accurate dynamic model of spatial manipulators that includes both link

and joint flexibility is developed. The algorithm is developed using the principle of virtual

work along with finite element method, and recursive kinematic formulation. The advantage

of general purpose algorithm is that it uses minimum set of equations that define the dynamics

of flexible manipulator, which is necessary for control design to reduce computational cost. In

addition, it also allows the dynamic modeling of any arbitrary manipulator configuration withrevolute joints.

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dynamic modeling and simulation of spatial manipulator with flexible links and joints

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